von Claes Johnson
Auszüge aus Kapitel 1:
Chapter 1: Main Objective
1.1 A Case Study of Mathematical Modeling
As a part of the Body&Soul Applied Mathematics educational program, we consider in this book the theory of special relativity proposed by Albert Einstein (1879-1955) in 1905. Our purpose is to exhibit fundamental aspects of mathematical modeling of the physical world we live in, through a careful study of special relativity. We choose to study special relativity because (i) relativity theory is supposed to form the foundation of modern physics, and (ii) relativity is not easy to understand. If we cannot understand the foundation, how can we understand what is built on it? We thus invite the reader to an experience of learning with the goal of understanding special relativity theory as a mathematical model of certain fundamental aspects of the World we live in. We start with open minds with a scientific attitude of only accepting what we can understand on good grounds, and not accepting anything by mere authority, and we will see where we end up. We promise that the reader will be surprised many times as we go along.
A mathematical model of some physical phenomenon builds on certain basic assumptions and derives by rules of logic and mathematical computation consequences of the basic assumptions, typically in the form of certain output from the model from certain input to the model. The basic assumption may be Newton’s 2nd Law with input consisting of the position and velocity of an object at an initial time combined with the force acting on the object and its mass, and the output may be the position and velocity at a later time.
From Newton’s law of gravitation
F = (G m1m2)/r²
where F is the gravitational force between two bodies of mass m1 and m2 at distance r and G is the gravitational constant (~ 9:81 meter per second²), you can e.g. predict by mathematical computation the coming position of the planets in our Solar system from their current positions and velocities.
But there is little hook: You also have to put in as data the mass of each planet and the Sun (and the gravitational constant G). And how do you determine the masses of the planets, when you cannot put them on a scale?
Nevertheless, Newton’s theory of gravitation became an immense success, which boosted mathematics and science based on mathematical modelling forming the basis of both the industrial society and the information society of today. In this case study of mathematical modeling, we shall focus on the following questions:
_ What motivated the development of special relativity?
_ What are the basic assumptions of special relativity?
_ What is the nature of these assumptions?
_ What are the basic consequences of these assumptions?
_ How can we test if the basic assumptions are valid?
_ Is there an alternative to special relativity?
Siehe auch vom Autor in diesem Blog:
- 21. Januar 2013